Power of Random Variable with Continuous Uniform Distribution has Beta Distribution

Theorem

Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the continuous uniform distribution on $\closedint 0 1$.

Let $n$ be a positive real number.

Then:

$X^n \sim \BetaDist {\dfrac 1 n} 1$

where $\operatorname {Beta}$ is the beta distribution.

Proof

Let:

$Y \sim \BetaDist {\dfrac 1 n} 1$

We aim to show that:

$\map \Pr {Y < x^n} = \map \Pr {X < x}$

for all $x \in \closedint 0 1$.

We have:

\(\ds \map \Pr {Y < x^n}\)

\(=\)

\(\ds \int_0^{x^n} \frac 1 {\map \Beta {\frac 1 n, 1} } u^{\frac 1 n - 1} \paren {1 - u}^{1 - 1} \rd u\)

Definition of Beta Distribution

\(\ds \)

\(=\)

\(\ds \frac {\map \Gamma {\frac 1 n + 1} } {\map \Gamma {\frac 1 n} \map \Gamma 1} \intlimits {n u^{\frac 1 n} } 0 {x^n}\)

Definition of Gamma Function, Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac n n x\)

Gamma Difference Equation

\(\ds \)

\(=\)

\(\ds x\)

\(\ds \)

\(=\)

\(\ds \frac 1 {1 - 0} \int_0^x \rd u\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \map \Pr {X < x}\)

Definition of Continuous Uniform Distribution

$\blacksquare$